Life Variant: 2/2 Life

Introduction | Object Counts | Still-lifes | P2 Oscillators | P4 Oscillators | Billiard table oscillators | Spaceships | Wick-stretchers



Introduction

The B2/S2 rule is similar to Life, except both birth and survival occur on exactly 2 living neighbors. This rule has the following interesting properties:


Object Counts

Computer searches have counted still-lifes up to 30 bits and period 2 oscillators up to 17 bits.

Numbers in bold face have been confirmed by computer search. Other given numbers are believed to be complete, but have not yet been verified. Lists with large numbers of objects that have not yet been counted are shown with "many".

Status of object lists on sub-pages is shown by background color:

Bits 23 45 67 89 1011 1213 1415 1617 1819 2021 2223 2425 2627 2929 30
Still-lifes 000 000 00 001 00 000 1 0 000 04 003 00 13
P2 oscillators 110 033 5215 2656 103250 4781129 many
P4 oscillators 0120 1240 2821 13many
Spaceships 001 10000 0002 111 111 114 7many



Still-lifes

Still-lifes are extremely rare in this rule. They are all highly artificial, and must employ extensive frameworks to induct all unwanted births on the outside. Since survival can only happen on 2, the only components permitted are L trominos and circular chains of bits (typically rounded rectangles, although at larger sizes, concave lakes are also possible). Since births can only occur on 2 neighbors, and such chains are bounded by empty cells with more than 2 everywhere except at outside corners, they act as stable immovable objects everywhere except at their outside corners. (As a side-effect, this permits construction of totally safe billiard table enclosures, since no matter what happens on the inside, the outside is safe from harm.)

Stable arrangements can be made of rectangular arrays of L trominos containing at least 2 in each dimension. Furthermore, one rectangular corner can be removed, yielding an L-shaped arrangement, as long as the two arms have width 2 or more. Thus, it is possible to create still-lifes of any population 3(xy+xj+iy) where x,y≥2 and i,j≥0. This simplifies to 3n for n in 4, 6, or any number 8 and larger.

A single pond can be stabilized by 16 trominos, allowing still-lifes of populations 56+3(3i+2j) where ij≥0, or 56+3n where n≥0.

Finally, a hollow box with 6 or more bits on the side can be stabilized by 3 L trominos on each corner, yielding still-lifes of populations 60+2n. If an additional L tromino is added on an inside corner, this yields still-lifes of populations 63+2n.

Thus, still-lifes are possible with populations 12, 18, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 56, 57, 60, and all sizes 62 and larger.

The following table shows all still-lifes up to 30 bits, plus a sampling of all other known sizes up to 65 bits. (Note that the number above each object indicates its population; none can be synthesized from gliders, and individual pattern files are not provided.)

B2/S2 still-lifes

Although not generally considered a still-life, the empty field (i.e. death) technically fulfills all the criteria, and it warrants being mentioned for several reasons. First, it is extremely rare for any patterns to totally die out. Second, unlike any of the other still lifes, it can be synthesized from gliders. Third, there are only three collisions of two gliders that don't expand infinitely, and one of them results in death.


Period-2 Oscillators

These are the 56 period 2 oscillators up to 11 bits, plus two larger ones for which syntheses are known (and that are both infinitely extensible).

B2/S2 period 2 oscillators

Blinker (2.1) [6] Yoyo (3.1) [8] 6.1 [x] 6.2 [x] 6.3 [x] 7.1 [x] 7.2 [x] 7.3 [x] 8.1 [x] 8.2 [x]
8.3 [x] 8.4 [x] 8.5 [x] 9.1 [x] 9.2 [x] 10.1 [x] 10.2 [x] 10.3 [x] 10.4 [x] 10.5 [x]
10.6 [x] 10.7 [x] Lines (10.8) [2] 10.9 [x] 10.10 [x] 10.11 [x] 10.12 [x] 10.13 [x] 10.14 [x] 10.15 [x]
11.1 [x] 11.2 [x] 11.3 [x] 11.4 [x] 11.5 [x] 11.6 [x] 11.7 [x] 11.8 [x] 11.9 [x] 11.10 [x]
11.11 [x] 11.12 [x] 11.13 [x] 11.14 [x] 11.15 [x] 11.16 [x] 11.17 [x] 11.18 [x] 11.19 [x] 11.20 [x]
11.21 [x] 11.22 [x] 11.23 [x] 11.24 [x] 11.25 [x] 11.26 [x] Three lines [4] Long lines [4]


Period-4 Oscillators

Other than the numerous period 2 oscillators, and the eccentric large billiard tables, the only other known oscillators are three small period 4 pulsators, plus their trivial extensions. Ones marked * are infinitely extensible. All up to 10 bits are shown, plus a few larger ones.

B2/S2 period 4 oscillators

V [5] Crown [10] Slash [10] V on v* [10] Crown on v* [15] Slash on v* [15]
Siamese crowns* [x] Crown on slash* [x] Crown on crown* [x] Slash on slash* [12] Trans v on v on v* [15] Cis v on v on v* [15]
Trans slash on slash on slash* [22] Cis slash on slash on slash* [x]  

Billiard Table Oscillators

Since still-lifes rely on heavily inducting the outsides of interiors that are inherently unstable, the same mechanisms used to construct them can also construct a variety of billiard table oscillators. A few small ones work with only L tromino stabilizations, but most work safely within a totally enclosed box. The period 5 oscillator shown below is infinitely extensible; a domino placed 5 or more bits away from both edges of a channel is a bi-directional period 5 glider gun that shoots gliders into the walls at both ends. (None of these can be synthesized from gliders, and individual pattern files are not provided.)

B2/S2 billiard tables

The last example shown is not really an oscillator at all. It was originally believed to be an oscillator with a period in excess of 20000, but was finally analyzed by Golly, using its HashLife algorithm, and the interior stablizes into a single blinker after 2,878,904 generations. It is a good illustration of the total robustness of billiard table interiors in this rule.


Spaceships

There is one basic spaceship, that has a period of 1, and moves with a velocity of c. Most variations of it produce incredibly dirty puffers with even periods, although there are a few that are clean spaceships and wick stretchers. Spaceships of any period 2n can be constructed by extending the examples below.

Two other spaceships have been found with velocities slower than c but these are very rare in this rule.

B2/S2 spaceships

P4 Escorted domino Spaceship [x] P8 Escorted domino Spaceship [x] c/2 Glider 21193 [x]
P2 Spaceship [x] P2 Escorted domino Spaceship [x]
P1 Glider; Glider 230 [4] c/3 Glider 9588 [x]


Wick-stretchers

The basic c spaceship can cleanly extend lines perpendicular to its direction of travel. This makes construction of wick-stretchers easy.

The tube-stretchers are special wick-stretchers that extrude an ever-expanding tube similar to a billiard table. Anything can be safely put inside, it cannot escape; the bottom and sides are impervious, and the top is receding at c, so nothing can catch up to it. The rightmost example shows how a single extra bit can forment total chaos, but in this case, the chaos is contained within a constrained area.

B2/S2 wick-stretchers

Two-line wick- stretcher [x] Blinkers on line wick- stretcher [x] Empty tube- stretcher [x] Glider- filled tube- stretcher [x] Chaos- filled tube- stretcher [x]



Other rules: B3/S23 (Conway's Life), Multi-colored Life, B2/S2 (2/2 Life), B34/S34 (3/4 Life), Niemiec's Rules.

See also: definitions, structure, search methodologies, other rules, news, credits, links, site map, search, expanded search, search help, downloads.

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This page was last updated on 2015-02-19.